Abstract—A traditional approach to segmentation of

magnetic resonance (MR) images is the Fuzzy C-Means (FCM)

clustering algorithm. However, the conventionally standard

FCM algorithm is sensitive to noise. To overcome the above

problem, a modified FCM algorithm (called MS-FCM later)

for MRI brain image segmentation is presented in this paper.

The algorithm is realized by incorporating the spatial

neighborhood information into the standard FCM algorithm

and modifying the membership weighting of each cluster. In

The proposed algorithm every point of the data set has a weight

in relation to every cluster. Therefore this weight permits to

have a better classification especially in the case of noise data.

The proposed algorithm is applied to both artificial synthesized

image and real image. Segmentation results demonstrate that

the presented algorithm performs more robust to noise than the

standard FCM algorithm.

Index Terms—Fuzzy C-Means, spatial information, image

segmentation, membership weighting.

I. INTRODUCTION

Segmentation of brain tissues in MRI (Magnetic

Resonance Imaging) images plays a crucial role in

three-dimensional volume visualization, quantitative morph

metric analysis and structure-function mapping for both

scientific and clinical investigations. Medical imaging

provides effective and non-invasive mapping of the anatomy

of subjects. Common medical imaging modalities include X-

ray, CT, ultrasound, and MRI. Medical imaging analysis is

usually applied in one of two capacities: a) to gain scientific

knowledge of diseases and their effect on anatomical

structure in vivo, and b) as a component for diagnostics and

treatment planning. MRI provides detailed images of tissues

and is used for both human brain and body studies. Data

obtained from MR images is used for detecting tissue

deformities such as cancers and injuries [1]. It aims to

partition an image into a set of non-overlapping regions

whose union is the original image.

FCM clustering algorithm, an unsupervised clustering

technique, has been successfully used for image

segmentation [2], [3]. Compared with hard C-Means

algorithm [4], FCM is able to preserve more information

from the original image. Its advantages include a

straightforward implementation, fairly robust behavior,

applicability to multichannel data, and the ability to model

Manuscript received June 18, 2012; revised August 5, 2012.

H. Shamsi is with the Department of Electrical and Computer Engineering,

University of Ataturk, Turkey (e-mail: Hamed.shamsi@atauni.edu.tr).

H. Seyedarabi is with the Department of Electrical and Computer

Engineering, University of Tabriz, Tabriz, Iran (e-mail:

Seyedarabi@tabrizu.ac.ir).

uncertainty within the data. A major disadvantage of its use

in imaging applications, however, is that FCM does not

incorporate information about spatial context, causing it to be

sensitive to noise and other imaging artifacts. The pixels on

an image are highly correlated, i.e. the pixels in the

immediate neighborhood possess nearly the same feature

data. Therefore, the spatial relationship of neighboring pixels

is an important characteristic that can be of great aid in

imaging segmentation. The spatial function is the weighted

summation of the membership function in the neighborhood

of each pixel under consideration. However, the standard

FCM does not take into account spatial information, which

makes it very sensitive to noise. In a standard FCM technique,

a noisy pixel is wrongly classified because of its abnormal

feature data.

This paper introduces a modified segmentation algorithm

for FCM clustering by incorporating spatial information and

altering the membership weighting of each cluster with

Fuzziness weighting exponent. The proposed algorithm

greatly attenuates the effect of noise and biases the algorithm

toward homogeneous clustering.

The organization of the paper is as follows. In Section 2,

traditional fuzzy c-means algorithm and spatial fuzzy

c-means are introduced. In Section 3, we obtain the fuzzy

c-means cluster segmentation algorithm based on modified

membership and modified cluster center. The experimental

comparisons are presented in Section 4. Finally, in Section 5,

we conclude and address the future work.

II. FUZZY C-MEANS

A. Traditional Fuzzy C-Means

The segmentation of imaging data involves partitioning

the image space into different cluster regions with similar

intensity image values. The most medical images always

present overlapping gray-scale intensities for different tissues.

Therefore, fuzzy clustering methods are particularly suitable

for the segmentation of medical images. There are several

FCM clustering applications in the MRI segmentation of the

brain. The Fuzzy c-means (FCM) can be seen as the

fuzzified version of the k-means algorithm. It is a method of

clustering which allows one piece of data to belong to two or

more clusters. This method (developed by Dunn [5] and

Modified by Bezdek [6]) is frequently used in pattern

recognition. The algorithm is an iterative clustering method

that produces an optimal c partition by minimizing the

weighted within group sum of squared error objective

function JFCM:

A Modified Fuzzy C-Means Clustering with Spatial

Information for Image Segmentation

Hamed Shamsi and Hadi Seyedarabi, Member, IACSIT

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International Journal of Computer Theory and Engineering, Vol. 4, No. 5, October 2012

N

j

c

i

ij

m

ijFCM vxuJ1

1

2 (1)

where X = {x1, x2,..., xn} pR is the data set in the

p-dimensional vector space, p is the number of data items, c

is the number of clusters with 2 ≤ c ≤ n-1. V = {v1, v2,..., vc} is

the c centers or prototypes of the clusters, vi is the

p-dimension center of the cluster i. U = {μij} represents a

fuzzy partition matrix with uij = ui (xj) is the degree of

membership of xj in the ith cluster, xj is the jth of

p-dimensional measured data. The fuzzy partition matrix

satisfies:

jiU

nju

cinu

ij

c

j

ij

n

j

ij

,,10

,.......,1,1

,.......,1,0

1

1

(2)

The parameter m is a weighting exponent on each fuzzy

membership and determines the amount of fuzziness of the

resulting classification; it is a fixed number greater than one.

The objective function JFCM can be minimized under the

Constraint of U. specifically, taking of JFCM with respect to

uij and vi and zeroing then respectively, tow necessary but

not sufficient conditions for JFCM to be at its local extreme

will be as the following:

njforandcifor

vx

vxU

c

k

m

kj

ij

ij

,......,2,1,.....2,1

)(1

1

2

(3)

ciforu

xuV

n

j

m

ij

n

j j

m

ij

i ,.....2,1)(

)(

1

1

(4)

Although FCM is a very useful clustering method, its

memberships do not always correspond well to the degree of

belonging of the data, and may be inaccurate in a noisy

environment, because the real data unavoidably involves

some noises.

B. Spatial Fuzzy C-Means (SFCM)

One of the important characteristics of an image is that

neighboring pixels have similar feature values, and the

probability that they belong to the same cluster is great. The

spatial information is important in clustering, but it is not

utilized in a standard FCM algorithm [7]. To exploit the

spatial information, a spatial function is defined as:

)( jxNBk ikij US (5)

The spatial function is the weighted summation of the

membership function in the neighborhood of each pixel

under consideration. Just like the membership function, the

spatial function sij represents the probability that pixel xj

belongs to ith clustering. The spatial function is the largest if

all of its neighborhood pixels belong to ith clustering, and is

the smallest if none of its neighborhood pixels belong to ith

clustering. The spatial function is incorporated into

membership function as follows:

njandcifor

c

k

qkj

Spkj

U

qijS

pijU

Uij

,.....,2,1,....,2,1

1

*

**

(6)

where p and q are parameters to control the relative

importance of both functions. In a homogenous region, the

spatial functions simply fortify the original membership, and

the clustering result remains unchanged. However, for a

noisy pixel, this formula reduces the weighting of a noisy

cluster by the labels of its neighboring pixels. As a result,

misclassified pixels from noisy regions or spurious blobs can

easily be corrected.

There are two steps at each clustering iteration. The first

step is to calculate the membership function in the spectral

domain and the second step is to map the membership

information of each pixel to the spatial domain and then

compute the spatial function from that.

III. A MODIFIED FUZZY C-MEANS CLUSTERING WITH

SPATIAL INFORMATION

We can compare the membership of central pixel with the

one of neighbor pixels in a window to analysis whether the

central pixel is classified rightly or not. This spatial

relationship is important in clustering; therefore a new spatial

function is defined as:

)(

1 )(

)(

2

1j

j

j

xHk c

t xHk

tk

xHk

kik

kikij

U

U

US

(7)

where H(xj) represents a square window centered on pixel xj

in the spatial domain. Introduced new spatial function has

two parts. The first part is controlled by 1k coefficient

caused that misclassified pixels from noisy regions can be

easily corrected. The second part is controlled by 2k

coefficient caused membership function quantitative

according to distance between pixels.

)exp(1

1

1

1kj

k

(8)

)exp(1

1

2

2

kj

kxx

(9)

The spatial function is incorporated into membership

function as follows:

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International Journal of Computer Theory and Engineering, Vol. 4, No. 5, October 2012

njandcifor

sU

sUU

c

k

q

ijp

kj

q

ijp

ij

ij

,.....,2,1,....,2,1

*

*

1

*

(10)

The choice of an appropriate objective function is the key

to the success of the cluster analysis and to obtain better

quality clustering results; so the clustering optimization is

based on objective function [8]. To meet a suitable objective

function, we started from the following set of requirements:

The distance between clusters and the data points assigned to

them should be minimized and the distance between clusters

should to be maximized [9]. The attraction between data and

clusters is modeled by term (12); it is the formula of the

objective function. Wen-Liang Hung proposed a new

algorithm called Modified Suppressed Fuzzy c-means

(MS-FCM), which significantly ameliorates the

performance of FCM due to a prototype-driven learning of

parameter α [10]. The learning process of α is based on an

exponential separation strength between clusters and is

updated at each iteration. The formula of this parameter is:

)minexp(

2

ki vv (11)

where β is a normalized term so that we choose β as a sample

variance. That is, we define β:

n

x

xwheren

xxn

j

j

n

j

j

11

2

(12)

But the remark which must be mentioned here is the

common value used for this parameter by all the data at each

iteration, which may induce in error. We propose a new

parameter which suppresses this common value of and

replaces it by a new parameter like a weight to each vector.

Or every point of the data set has a weight in relation to every

cluster. Therefore this weight permits to have a better

classification especially in the case of noise data. So the

weight is calculated as follows:

)exp(1

1

1

2

2

n

j

j

ij

ji

ncvx

vxW

(13)

where wji is weight of the point j in relation to the class i. This

weight is used to modify the fuzzy and typical partition.

The objective function of the MS-FCM can be formulated

as follows:

2

1 1

)( ik

n

k

c

i

m

ji

m

ikmsfcm vxWuJ

(14)

Modified FCM approach is given below:

Step 1: Select the data set.

Step 2: Fix m >1 and 12 nc and give c

initial cluster centers Vi.

Step 3: Compute Uij with Vi by Eq. (3).

Step 4: Compute 1k and 2k by Eq. (8) and (9).

Step 5: Compute ijS and jiW by Eq. (7) and (13).

Step 6: Update the membership matrices by Eq. (10),

Update the centroids using (4).

Step 7: if oldnew VV Stop the iteration

otherwise, go to step 4.

IV. EXPERIMENTAL RESULTS

In order to verify the effectiveness of the proposed

algorithm, we give some experiments using both artificial

synthesized image and real image to compare the

performance of the proposed algorithm with that of the

standard FCM algorithm.

A. Experimental Results on Synthetic Image

Fig. 1a shows a synthesized image with three classes, the

image degraded by the Salt-Pepper noise (noisy density d =

0.02) and Gaussian noise (u = 0, δ = 0.02) is shown in Fig. 1b

and 1c, respectively.

(a) (b) (c)

Fig. 1. Synthesized image: (a) original image, (b) image degraded by

Salt-Pepper noise, (c) image degraded by Gaussian noise.

Figs. 2a-2b-2c display the clustering results of the

Salt-Pepper degraded image using the FCM, SFCM and

MS-FCM algorithm, correspondingly, the clustering results

on Gaussian degraded image were shown in Figs. 3a-3b-3c.

(a) (b) (c)

Fig. 2. Comparison of segmentation results on synthetic image which is

corrupted by 2% salt- pepper noise. (a) FCM result, (b) SFCM result (c)

MS-FCM result.

(a) (b) (c)

Fig. 3. Comparison of segmentation results on synthetic image which is

corrupted by Gaussian noise (u = 0, δ = 0.02) (a) FCM result (b) SFCM result

(c) MS-FCM result.

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International Journal of Computer Theory and Engineering, Vol. 4, No. 5, October 2012

In order to obtain a quantitative comparison, two types of

cluster validity functions, fuzzy partition and feature

structure, are often used to evaluate the performance of

clustering in different clustering methods. The representative

functions for the fuzzy partition are partition coefficient Vpc

[11] and partition entropy Vpe [12]. They are defined as

follows:

)(1

)(V1 1

2

pc

n

K

c

i

ikun

U (15)

)log(1

)(V1 1

pe

n

K

c

i

ikik uun

U (16)

The idea of these validity functions is that the partition

with fuzziness means better performance. As a result, the best

clustering is achieved when the value Vpc is maximal or Vpe

is minimal.

Table I tabulates the Vpc and Vpe and number of iteration

of the three algorithms on two different noise degraded

images shown in Figs.1b and 1c, respectively.

TABLE I: COMPRESSION OF THE CLUSTERING RESULTS ON TWO KIND NOISE

DEGRADED SYNTHETIC IMAGE USING THREE FCM ALGORITHMS

Noise type Algorithm Vpc Vpe iter

Salt-pepper FCM 0.9961 0.0049 12

Salt-pepper SFCM 0.9984 0.0016 6

Salt-pepper MS-FCM 0.9991 0.0008 5

Gaussian FCM 0.8308 0.1386 43

Gaussian SFCM 0.9630 0.0323 11

Gaussian MS-FCM 0.9964 0.0042 6

From Table I, obviously, MS-FCM achieves better

performance than FCM and SFCM, which demonstrates the

modified SFCM algorithm (MS-FCM) a visually significant

improvement of robustness to noise over the FCM and SFCM

algorithm.

B. Experimental Result on Real Image

Fig.4a shows an MRI brain image with 256*256 pixels.

The image degraded by the Gaussian noise (u = 0, δ = 0.02)

and the Salt-Pepper noise with noisy density d =0.02 is

shown in Fig. 4b and 4c, respectively.

(a) (b) (c)

Fig. 4. MRI image: (a) original image, (b) image degraded by Salt-Pepper.

Figs. 5a-5b-5c display the clustering results of the

Salt-Pepper degraded image using the FCM and SFCM and

MS-FCM algorithm, correspondingly, the clustering results

on Gaussian degraded image were shown in Figs. 6a-6b-6c.

Visually, the comparison of segmentation results indicates

that MS-FCM algorithm greatly reduces the effect of noise.

Table II summarizes the partition coefficient Vpc and the

partition entropy Vpe of the two algorithms on the two

different degraded images.

(a) (b) (c)

Fig. 5. Comparison of segmentation results on MRI image which is corrupted

by 2% Salt- Pepper noise. (a) FCM result (b) SFCM result (c) MS-FCM

result.

(a) (b) (c)

Fig. 6. Comparison of segmentation results on MRI image which is corrupted

by Gaussian noise (u = 0, δ = 0.02) (a) FCM result (b) SFCM result (c)

MS-FCM result.

TABLE II: COMPRESSION OF THE CLUSTERING RESULTS ON TWO KIND

NOISE DEGRADED MRI IMAGE USING THREE FCM ALGORITHMS.

Noise type Algorithm Vpc Vpe iter

Salt-pepper FCM 0.8631 0.1081 81

Salt-pepper SFCM 0.8647 0.1066 79

Salt-pepper MS-FCM 0.9130 0.0628 52

Gaussian FCM 0.7669 0.1812 77

Gaussian SFCM 0.7901 0.1573 71

Gaussian MS-FCM 0.8359 0.1203 42

Table II shows that our proposed algorithm improves

significantly the performances of clustering both Gaussian

degraded image and Salt-Pepper degraded image compare to

the standard FCM algorithm.

V. CONCLUSION

The main drawback of the standard FCM for image

segmentation is that the objective function does not take into

consideration the spatial information in the image, but deal

with images as the same as separate points. Therefore, as

mentioned in many literatures [13, 14] the standard FCM

algorithm is sensitive to noise and a noisy pixel is always

wrongly classified because of its abnormal feature.

In this paper, we proposed a new modify spatial FCM that

incorporates the spatial information into the membership

function to improve the segmentation results. In the new

spatial function we used two contribution factors. The first

one was according to distances between central pixels with

neighbor pixels. The second factor was calculated according

to value difference of central pixel with neighbor pixels.

Using of these contribution factors caused that spatial

function is made of according to distance and value pixels.

The new method was tested on MRI images and evaluated

by using various cluster validity functions. Preliminary

results showed that the effect of noise in segmentation was

considerably less with the new algorithm than with the

conventional FCM.

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Hamed Shamsi received the B.Sc. degrees from

Islamic Azad University of Urumiyeh in 2008 and the

M.Sc. degrees from Islamic Azad University of

Najafabad Branch in 2010 all in Electrical

Engineering field (telecommunication). He is

currently Ph.D. student in Ataturk University, turkey.

His research interests are image processing, image

segmentation, medical signal processing, pattern

recognition, and fuzzy logic.

Hadi Seyedarabi Received B.S. degree from

University of Tabriz, Iran, in 1993, the M.S. degree

from K.N.T. University of technology, Tehran, Iran in

1996 and Ph.D. degree from University of Tabriz,

Iran, in 2006 all in Electrical Engineering. He is

currently an assistant professor of Faculty of

Electrical and Computer Engineering in University of

Tabriz, Tabriz, Iran. His research interests are image

processing, computer vision, Human-Computer

Interaction, facial expression recognition and facial

animation.

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